Singular solutions in problems of optimal control

A particular class of optimization problems, in which the system equations and index of performance are linear in the control variable, is examined in detail. Pontryagin\´s Maximum Principle seems to indicate an optimal control of the bang-bang type for this class of problems. However, it is shown that the optimal control may actually consist of intervals of variable control effort (called "singular control") combined with intervals of bang-bang control. The conditions which characterize singular control are derived. Some techniques are given which may be helpful in detecting and calculating singular controls in this class of problems. In general, it cannot be stated a priori that singular control will necessarily constitute part of the optimal control. Two examples are worked out in detail to illustrate application of the techniques given in the paper.